Why is a vertex from another island is moving if I move an UV island?
Two possible settings: 


As @moonboots has suggested, you may have 'Keep UV And Edit Mode Mesh Selection In Sync' checked, in which case the whole UV is made available in the UV window, and you're selecting mesh vertices, so all UV face-corners associated with the vertices are selected:





Or you may not have Sync Selection checked, in which case you are selecting UV face-corners directly, but you have the option via the 'Sticky Selection Mode'  to select them separately, or together if they share a UV location, or together if they share a vertex in the mesh: